↑

Properties

Label 2-1176-168.131-c0-0-4
Degree $2$
Conductor $1176$
Sign $0.0587 + 0.998i$
Analytic cond. $0.586900$
Root an. cond. $0.766094$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.608 − 0.793i)3-s + (0.499 + 0.866i)4-s + (−0.923 + 0.382i)6-s − 0.999i·8-s + (−0.258 − 0.965i)9-s + (1.22 − 0.707i)11-s + (0.991 + 0.130i)12-s + (−0.5 + 0.866i)16-s + (0.923 + 1.60i)17-s + (−0.258 + 0.965i)18-s + (−0.662 − 0.382i)19-s − 1.41·22-s + (−0.793 − 0.608i)24-s + (−0.5 − 0.866i)25-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.608 − 0.793i)3-s + (0.499 + 0.866i)4-s + (−0.923 + 0.382i)6-s − 0.999i·8-s + (−0.258 − 0.965i)9-s + (1.22 − 0.707i)11-s + (0.991 + 0.130i)12-s + (−0.5 + 0.866i)16-s + (0.923 + 1.60i)17-s + (−0.258 + 0.965i)18-s + (−0.662 − 0.382i)19-s − 1.41·22-s + (−0.793 − 0.608i)24-s + (−0.5 − 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0587 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0587 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $0.0587 + 0.998i$
Analytic conductor: \(0.586900\)
Root analytic conductor: \(0.766094\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (803, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :0),\ 0.0587 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9097752647\)
\(L(\frac12)\) \(\approx\) \(0.9097752647\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (-0.608 + 0.793i)T \)
7 \( 1 \)
good5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (-0.923 - 1.60i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.662 + 0.382i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + 0.765T + T^{2} \)
43 \( 1 - 1.41T + T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.923 + 1.60i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1.60 - 0.923i)T + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 - 1.84T + T^{2} \)
89 \( 1 + (0.382 - 0.662i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 - 0.765iT - T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.619879821741283218589432594228, −8.827482631962256422983829410907, −8.304872767997201154019323404181, −7.59892421256629673595525207998, −6.52167759942808918936885809730, −6.08321256483770650211729700321, −4.04695561615866462684625347046, −3.38668287123300065574059830769, −2.16241836282105001518672740879, −1.15286029227229812318135210349, 1.64067791275662581257882755010, 2.91046876619753754663602804707, 4.15616338800707896379110565196, 5.10508239764047351405783531820, 6.05055735438629240002922699890, 7.20777877028670573731453550555, 7.68218441443449954715954625232, 8.795163111687841242515593458586, 9.308255567466078695913678941634, 9.861661720625089785547745913858

Graph of the $Z$-function along the critical line